The constant \(\lambda \)  above is called the dilatation constant for the \((q,p)\) -Poincaré inequality.
The \((1,p)\) -Poincaré inequality
for \(1\le p<Q\) , where \(Q\)  is as in (REF ),
together with the doubling condition (REF ) implies a \((q,p)\) -Poincaré inequality
with \(q=Qp/(Q-p)\) ; see [1]}, [2]}.
